1896 Arrhenius (1119300), страница 6
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From this cause both the radiation from airto space (/~10' in eq. 1) and also the radiation of the earthto the air (fi~lv(T~--6 ') in eq. 2), are greatly reduced, andthe air has a much greater effect as protecting against theloss of heat to space than is assumed in these equations, andconsequently also in eq. (3). If we knew the difference oftemperature between the two layers of the air that radiate tospace and absorb the earth's radiation~ it would be easy tointroduce the necessary correction in formui~e (1), (2), and3). For this purpose I have adduced the following consideration.As at the mean composition of the atmosphere ( K = I ~260Prof.
S. Arrhenius on the Influence of Carbonlc AcidW = l ) about 80 p. c. of the earth's radiation is absorbed inthe air, we may as mean temperature of the absorbing layerchoose the temperature at the height where 40 p. c. of theheat is absorbed. Since emission and absorption followthe same quantitative laws, we may as mean temperature ofthe emitting layer choose the ~emperature at the height whereradiation entering from space in the opposite direction to theactual emission is absorbed to the extent of 40 p. c.Langley has made four measurements of the absorptivepower of water-vapour for radiation from a hot Leslie cubeof 100 ° C.* These give nearly the same absorption-coefficient if Pouillet's formula is used for the calculation.
Fromthese numbers we calculate that for the absorption of 40 p. c.of the radiation it would be necessary to intercalate so muchwater-vapour between radiator and bolometer thaff, whencondensed, it would form a layer of water 3"05 millimetresthick. If we now suppose as mean for the whole earth K = Iand W = 1 (see Table VI.), we find that vertical rays from theearth, if it were at 100 °, must traverse 305 metres of air tolose 40 p.c. Now the earth is only at 15 ° C., but this cannotmake any great difference. Since the radiation emanates in alldirections, we have to divide 305 by 1"61 and get in this way209 metres.
In consequence of the lowering of the quantityof water-vapour with the height ~ we must apply a slightcorrection, so that the final result is 233 metres. Of coursethis number is a mean value, and higher values will holdgood for colder~ lower for warmer parts of the earth. In sosmall a distance from the earth, then, 40 p. c. of the earth'sradiation should be stopped. Now it is not wholly correct tocalculate with Pouillet's formula (it is rather strange thatLangley's figures agree so well with it), which gives necessarily too low values.
But, on the other hand, we have notat all considered the absorption by the carbonic acid in thispart, and this may compensate for the error mentioned. Inthe highest layers of the atmosphere there is very little watervapour, so that we must calculate with carbonic acid asthe chief absorbent. From a measurement by ~ngstrSm:~,we learn that the absorption-coefficients of ~vater-vapour andof carbonic acid in eqlml quantities (equal number of molecules)are in the proportion 8 1 : 6 2 . This ratio is valid for theleast hot radiator that/~ngstrgm used, and there is no doubt* Langley, "Tem!0erature of the ~$oon,""o 186H~nn, Meteorolog~schexn p.~196(1894).t..
ZeztscI~q/t,..••:[: AngstrSm, ~ihang till I(. Vet.-Ak. ttandl. Bd. xv. Afd. 1, No. 9~pp. 11 and 18 (1889).ia the Air upon t]~e Teml)erature of the G~'ound.26[that the radiation of the earth is much less refrangible. ]~utin the absence of a more appropriate determination we mayuse this for our purpose ; it is probable that for a less hotradiator the absorptive power of the carbonic acid wouldc o m e o u t a little g r e a t e r c o m p a r e d w i t h t h a t of w a t e r - v a p o n r ,for the absorption-bands of C02 ar% on the whol% lessrefi'angible than those of H20 (see pp.
246-248). Using thenumber 0"03 vol. p. c. for the quantity of carbonic a(dd inthe atmosphere, we find that rays which emanate from theupper part of the air are derived to the extent of 40 p. c. froma layer that constitutes 0"145 part of the a|mosphere. Thiscorresponds to a height of about 15,000 metres. Concerningthis value we may make the same remark as on the foregoingvalue. In this case we have neglected the absorption by thesmall quantities of water-vapour in the higher atmosphere.The temperature-difference of these two layers--the one absorbing, the other radiating--is, according to Glaisher'smeasurements* (with a little extrapolation), about 42 ° C.For the clouds we get naturally slightly modified numbers.We ought to take the mean height of the clouds that areilluminated by the sun.
As such clouds I have chosen thesummits of the cumuli that lie at an average height of1855 metres, with a maximum height of 3611 metres and aminimum of 900 metres ¢. I have made calculations formean values of 2000 and 4000 metres (corresponding to differences of temperature of 30 ° C. and 20 ° C. instead of 42 ° (~.for the earth's surface).If we now wish to adjust our formulm (1) to (3), we havein (1) and (2) to introduce 0 as the mean temperature of theradiating layer and (0 + 42), (0 + 30), or (~ + 20)respectivelyfor the mean temperature of the absorbing layer.
In thefirst case we should use v = l and v=0"925 respectively, inthe second and the third case v=0"22.We then find instead of the formula (3)T 4--Kl -~ v ( l -- fl) 'another very similar formulaT4=Ki + er(1--fl)' . . . . . .(4)* Joh. ]Kiiller's Zehr~ueh d. kosmisehen _Physik, 5 t~ AUfl. p. 539(Braunschweig, 1894).J- According to the measurements of ]~kliolm and ttagstrom, Bihangtill K, Sv, Vet-Ak.
ttandhngar, Bd. xii. Afd. 1, No. 10, t~. 11 (1886).262Prof. S. Arrhenius on the Influence of Carbonic Acidwhere e is a constant with the values 1"88, 1"58, and 1"37respectively for the three cases *. In this way we find thefollowing corrected values which represent the variation oftemperature, if the solid ground changes its temperature l ° C.in consequence of a variation of fl as calculated by means offormula (3).TABLE V.--Correction Factors for the Radiation.Solid Water, Snow, ~louds(v= 0"22)at aheight o~ground, u=0'925./ v=0"5.
0 m. 2000m. 4000m.0750"850-951.00-T.5-U1"601 "691"81l '88rTU/-7.9--U - - -0"49-1'521-591"681-740"950"950-940"940"470'460-430"410"420"400"380"360'350"370'350'330"310"30I f we now assume as a mean ibr the whole earth K = I andW-----l, we get fl=0"785, and taking the cloudcd part to be52"5 p. c. and the clouds to have a height of 2000 metres,further assuming the unclouded remainder of the earth'ssurface to consist equally of land and water, we find as averagevariation of temperature1"63 × 0"2385 + 1"54 x 0"2385 + 0'39 × 0"525 = 0 ' 9 7 9 ,or very nearly the same effect as we may calculate directlyfrom the formula (3).
On this ground I have used thesimpler formula.]n the foregoing I have remarked that according to myestimation the air is less transparent for dark heat than onLangley's estimate and nearly in the proportion 3 7 " 2 : 4 4 .How great an influence this difference may exercise is veryeasily calculated with the help of formula (3) or (4). According to Langley's valuation, the effect should be nearly15 p. c. greater than according to mine. l~ow I think that m )estimate agrees better with tim great absorption tha~ Langleyhas found for heat from terrestrial radiating bodies (see p.
260),and in all circumstances I have preferred to slightly underestimate than to overrate the effect in question.• 1.ss={288~4 1"58=/276~4and1•37 / \2t',6 446oabsolute temperature of the higher radiating layer of the air,the meanin the Air upon the Temperature of the Ground.263IV. Calculation of the Variation of Temperature that wouldensue in consequence of a given Variation of the CarbonicAcid in the Air.We now possess all the necessary data for an estimation ofthe effect on the earth's temperature which would be theresult of a given variation of the a~rial carbonic acid. Weonly need to determine the absorption-coefficient for a certainplace with the help of Table IfI.
if' we know the quantity ofcarbonic acid ( K = I now) and water-vapom" (W) of thisplace. By the aid of Table IV. we at first determine the factorp that gives the mean path of the radiation fi'om the earththrough the air and multiply the given K- and W-values bythis factor. Then we determine the value of/3 which corresponds to pK and pW. Suppose now that the carbonic acidhad another concentration K] (e.g.
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